By [1]} with \(t=1\) , \(\Vert \nu \Vert ^2\geqslant c_*(A)\)  for all \(\nu \in \Gamma _A\) , and hence
\(\inf _{\nu \in \Gamma _A^+}\,\Vert \nu \Vert ^2\geqslant \inf _{\nu \in \Gamma _A}\,\Vert \nu \Vert ^2\geqslant c_*(A).\) 
It is thus enough to consider the case where \(c_*(A)<\infty \) .
The uniqueness of the solution to the minimum energy problem over \(\Gamma _A\)  \((\Gamma _A^+)\)  follows from the convexity of \(\Gamma _A\)  \((\Gamma _A^+)\) , which in turn is obvious from Lemma REF .
The (unique) measure \(\gamma \in \Gamma _A\)  of minimal energy in the class \(\Gamma _A\)  does exist, and moreover it satisfies (REF )–() (see [1]}). It thus remains to prove that the same \(\gamma \)  solves the minimum energy problem over \(\Gamma _A^+\) . In view of (REF ), this will follow once we show that this \(\gamma \)  is positive.
Assume \(c_*(A)>0\) , for if not, \(\gamma =\gamma _A=0\) .
Analyzing the proof of Theorem 4.1 in [1]}, we deduce the following.
Lemma 6.2 The above measure \(\gamma \)  is the strong limit of \((\gamma _{K_j})\) , where \((K_j)\)  is an increasing sequence of compact sets \(K_j\subset A\)  with \(c(K_j)>0\)  and such thatSuch a sequence \((K_j)\)  exists by virtue of (REF ).
\(\lim _{j\rightarrow \infty }\,c(K_j)=c_*(A),\) 
while \(\gamma _{K_j}=\lambda _{K_j}/w(K_j)\in \mathcal {E}^+_{K_j}\)  is the (unique) capacitary measure on \(K_j\) .

Since the kernel is perfect, \(\gamma _{K_j}\rightarrow \gamma \)  also vaguely. Hence \(\gamma \)  is indeed positive, the cone \(\mathfrak {M}^+\)  being vaguely closed [4]}.
